Let $\displaystyle E$ be a compact connected subset of the complex plane that contains more than one point. Show that the family of meromorphic functions on a domain $\displaystyle D$ that omits $\displaystyle E$ (that is, with range in $\displaystyle \mathbb{C}^* \backslash E)$ is a normal family of meromorphic functions.

The hint in the back of the book says to map $\displaystyle \mathbb{C}^* \backslash E$ conformally onto $\displaystyle \mathbb{D}$, apply thesis version of Montel's theorem. That version states:

Suppose $\displaystyle \mathcal{F}$ is a family of analytic functions on a domain $\displaystyle D$ such that $\displaystyle \mathcal{F}$ is uniformly bounded on each compact subset of $\displaystyle D$. Then every sequence in $\displaystyle \mathcal{F}$ has a subsequence that converges normally in $\displaystyle D$, that is, uniformly on each compact subset of $\displaystyle D$.

I am still confused on how to prove this. I would appreciate some help on this problem. Thanks.